U.S. patent number 7,056,185 [Application Number 10/958,005] was granted by the patent office on 2006-06-06 for single axle wireless remote controlled rover with omnidirectional wheels.
Invention is credited to Thomas Anagnostou.
United States Patent |
7,056,185 |
Anagnostou |
June 6, 2006 |
Single axle wireless remote controlled rover with omnidirectional
wheels
Abstract
A wireless remote controlled rover (100) having a circular outer
boundary when viewed in profile, comprising an elongated
round-profile frame (20) outfitted with a pair of omnidirectional
wheels (101L,R), rotatable in opposite directions by a drive system
(50), for steering. Each of omnidirectional wheels (101L,R)
comprises a set of secondary wheels (150L,R) akin to large diameter
hollow flexible shafts arcuately bent and distributed in a polar
array about corresponding hubs (82L,R). The secondary wheels
(150L,R) are rotatable by the drive system (50) for longitudinal
thrust of rover (100). A user controls the steering and thrust
actions via remote control (166) for navigation and to compensate
for incidental inertial rolling/careening effects of the two-wheel
round-profile rover (100) while in motion. An onboard intelligent
electronic processing unit (33) uses sensory feedback for
autopilot, or assistive navigation, during certain scenarios such
as performing user-requested optimum stopping maneuvers.
Inventors: |
Anagnostou; Thomas (Newark,
DE) |
Family
ID: |
36568838 |
Appl.
No.: |
10/958,005 |
Filed: |
October 4, 2004 |
Current U.S.
Class: |
446/456; 180/167;
180/218; 180/6.5; 180/7.1; 446/462 |
Current CPC
Class: |
A63H
17/262 (20130101); A63H 30/04 (20130101); B60K
7/0007 (20130101); B62D 61/00 (20130101); B60K
17/046 (20130101); B60K 2001/001 (20130101); B60K
2007/0046 (20130101); B60K 2007/0069 (20130101); B60L
2220/46 (20130101) |
Current International
Class: |
B62D
61/00 (20060101) |
Field of
Search: |
;180/218,7.1,7.2,167,6.2,6.5 ;446/431,443,454,456,457,462 |
References Cited
[Referenced By]
U.S. Patent Documents
Primary Examiner: Hurley; Kevin
Claims
What is claimed is:
1. A wireless-remote-controlled round profile axle-like rover
responsive to control signals from a remote transmitter unit,
comprising: a) a pair of ground engaging wheels all having a common
central axis of rotation, each of said wheels being omnidirectional
and comprising a hub rotatable about said central axis, said hub
having a group of peripheral circumrotatory ground-contact elements
circumferentially disposed thereof, each of said peripheral
elements having a respective spin axis independent in relation to
said central axis and each of said peripheral elements being fully
rotatable about its own respective said spin axis, said group of
peripheral elements defining a substantially circular
ground-contact boundary about said central axis on each of said
wheels, and b) a wireless remote controlled wheel drive engaging
said hub and engaging said group of peripheral elements on each of
said wheels and selectively distributing rotational energies among
said hub and said group of peripheral elements respectively on each
said wheels; whereby said rover substantially resembles a single
two-wheeled axle that can rollably careen unobstructed on a ground
surface at a speed substantially orthogonal to said central axis of
rotation, as well as steer by distribution of opposing rotational
energies to each of said hubs, and propel linearly along said
central axis by distribution of uniform rotational energies to said
groups of peripheral elements of said wheels.
2. The rover of claim 1 wherein said rover is structurally confined
within an imaginary cylindrical envelope defined by said central
axis and the radius of said ground-contact boundary of said
wheels.
3. The rover of claim 1 wherein said rover has a balanced weight
distribution about said central axis of rotation and has a center
of gravity substantially coinciding with said central axis of
rotation.
4. The rover of claim 1 wherein said wireless remote controlled
wheel drive includes mechanical or electronic constraints rendering
said hubs always rotatable in opposite directions relative to each
other and rendering each of said peripheral elements on both said
wheels always rotatable synchronously with uniform rotational speed
and direction.
5. The rover of claim 1 wherein each of said omnidirectional wheels
further comprises vivid exterior indicia mutually contrasting
between each of said omnidirectional wheels, whereby a human
operator can use as visual reference of orientation of said rover
from a distance.
6. The rover of claim 1 wherein: a) each of said hubs further
comprises a plurality of drive spindles tangentially and rotatably
disposed on the periphery of said hubs equidistantly about the hub
rotation axis and receiving rotational energy from said wireless
remote controlled wheel drive, and b) said group of peripheral
elements is a plurality of cylindrical flexible shafts, each bent
to a natural arc and each being endwise detachably attached between
pairs of consecutive said drive spindles, said flexible shafts
defining a substantially circular ground contact perimeter and
serving as secondary wheels; whereby each of said omnidirectional
wheels can roll on a ground surface as well as translate laterally,
at will, along the hub rotation axis relative to a ground
surface.
7. The rover of claim 1 wherein said wireless remote controlled
wheel drive comprises: a) a pair of drive motors, b) a drivetrain
engaging said motors and receiving output rotational energies from
said motors, said hubs engaging said drivetrain and receiving a
first range of rotational energies, said groups of peripheral
elements engaging said drivetrain and receiving a second range of
rotational energies, c) a processing unit communicatively connected
with each of said drive motors and controlling the angular
velocities of said motors, said processing unit in communication
with said remote transmitter unit, and d) a power source connected
to said motors and said processing unit, said motors, said
drivetrain, and said processing unit cooperate and render said hubs
always rotatable in opposite directions relative to each other and
render each of said peripheral elements always rotatable
synchronously with uniform angular velocities on both said wheels;
whereby said processing unit interprets user signals and regulates
accordingly the angular velocity of each of said motors and
consequently the portions of rotational energies distributed among
said hub and said group of peripheral elements respectively on each
of said wheels.
8. The rover of claim 7 further comprising one or more orientation
and velocity sensors communicatively connected with said processing
unit and providing sensory feedback; whereby said rover is made
self-aware of several orientation and velocity parameters such as
longitudinal inclination as well as speed and direction of
travel.
9. The rover of claim 7 further comprising: a) a hollow shaft
having at least one open end and having said hubs rotatably
disposed endwise thereof, said shaft defining an inner cylindrical
chamber and having a longitudinal axis coincidental with said
central axis, and b) at least one cap or closure lockably mounted
on each said open ends of said hollow shaft, said power source
disposed within said chamber; whereby said closure allows user
access to said chamber for replacement of said power source.
10. The rover of claim 9 wherein said drivetrain comprises: a) a
pair of planetary gearing systems, each engaging with a
corresponding one of said wheels, and each of said planetary
gearing systems comprising: i) a ring gear rotatably and coaxially
mounted on said shaft, ii) a sun gear rotatably mounted on said
shaft and coaxial with said ring gear, iii) a plurality of
planetary gears meshing with said ring gear and with said sun gear
and arranged in a polar array about the central axis of said ring
gear and said sun gear, and iv) a planetary gear carrier ring
having said planetary gears rotatably disposed thereof and said
carrier ring being fixedly secured on said hub of a corresponding
one of said wheels, b) a first transmission means connecting each
of said sun gears with a first of said drive motors, c) a second
transmission means connecting each of said ring gears with a second
of said drive motors, d) a plurality of drive spindles tangentially
and rotatably disposed equidistantly on the periphery of said hubs
of said wheels, and e) a plurality of spindle transmission means
coupling each of said planetary gears to a respective one of said
drive spindles on a corresponding said wheel; whereby said carrier
rings convey rotation on said hubs and said planetary gears convey
rotation on said flexible shafts and the relative rotational speeds
between the first and second of said drive motors determines the
proportion of rotational energies delivered to said hub and said
flexible shafts of each of said wheels.
11. A method of roving on a surface using a remote vehicle and a
wireless controller, comprising: a) providing a pair of coaxial
omnidirectional wheels rotatable relative to each other about a
central longitudinal axis, said wheels having hubs with peripheral
roller elements, said roller elements allowing said wheels to
propel laterally along their hub axis of rotation, said wheels can
also roll as a whole on a surface about their hub axis of rotation,
b) providing a wheel drive means for selectively rotating said hubs
always in opposite directions relative to each other so as to
enable directional changes of said vehicle, said wheel drive means
selectively rotating said roller elements of both said wheels
simultaneously with uniform rotational velocities so as to enable
linear travel of said vehicle along said longitudinal axis, c)
providing a transmitter unit for use remotely by a human operator,
said transmitter unit comprising steering and thrust controls, d)
providing a processing unit on said vehicle for interpreting input
from said transmitter unit and translating said input into control
signals for said wheel drive means, said processing unit
translating i) steering signals into rotation of said hubs in
opposite directions relative to each other so that said vehicle can
change direction of travel, and ii) thrust signals into
simultaneous rotation of said peripheral elements of both said
wheels at uniform rotational speeds and directions so that said
vehicle can travel linearly along said longitudinal axis.
12. The method of claim 11 further comprising: a) providing sensor
means on said vehicle conveying feedback input signals to said
processing unit for monitoring inertial velocity, rotational
velocity, axial orientation, and axial inclination, b) providing
deceleration controls for said transmitter unit, said processing
unit translating deceleration signals into a series of automated
steering actions orchestrated by said processing unit in
consultation with said sensor means, said processing unit ordering
said wheel drive into successive steering maneuvers, each said
maneuver being a swift steering attempt of bringing the vehicle
said longitudinal axis temporarily in line with the direction of
inertial roll velocity of said vehicle and then swiftly restoring
said longitudinal axis to its previous orientation and each said
swift steering attempt reducing the vehicle inertial velocity by a
small amount and repeating until the vehicle stops completely or
decelerates to a desired inertial roll velocity.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
Not Applicable
FEDERALLY SPONSORED RESEARCH
Not Applicable
SEQUENCE LISTING PROGRAM
Not Applicable
BACKGROUND OF THE INVENTION
The present invention relates to remote controlled vehicles,
particularly to uncrewed rovers, terrain probing exploration
vehicles, and recreational toy stunt vehicles, and specifically to
a two-wheeled rover with a pair of omnidirectional wheels coaxially
rotatable, having actively driven secondary wheels and a dynamic
stability electronic system.
Remote controlled vehicles have been introduced in the past for
terrain exploration and reconnaissance missions. A factor of
importance, pertinent to those scenarios, is the proper design of
the vehicles enabling them to negotiate unpredictable terrain
morphology.
Carriage-on-wheels type of rovers have an inherent vulnerability,
in that, unforeseen terrain factors may cause the rover to tumble,
or tip to its side, loosing wheel ground contact and be unable to
recover. Combinations of design parameters such as a low center of
gravity, long wheelbase and larger track width are usually applied
to minimize the tipping tendency. However these parameters are also
competing against favorable wheel clearance and overall dimensional
compactness; meaning that the vehicle body needs to be lower to the
ground and to occupy a larger area for added stability.
Several concepts have been brought forth in robotic applications
such as involving multi-limbed, multi-symmetrical vehicles
attempting to address issues of balanced, tip-resistant,
orientation agnostic designs. Multi-limbed robotic exploration
vehicles, although conceptually aspiring to simple geometric
shapes, tend to materialize as mechanically and electronically
complex structures of low speed potential.
Accordingly it would be beneficial to have a high speed, remote
controlled, structurally compact, orientation agnostic, exploration
rover that can be handled relatively carelessly with minimal
concern of it loosing traction, or becoming stranded by tipping
over, or not being able to maneuver between narrow passages.
On the recreational aspect, numerous embodiments of radio
controlled (r/c) surface roving toy and hobby vehicles have been
introduced in the past. Often these toy vehicles are scaled-down
incarnations of real life transportation equivalents. For example a
two wheel r/c toy is usually a miniature motorcycle. Similarly a
four wheel r/c toy is often a scaled-down version of a car, a truck
and the like. Furthermore, since transportation vehicles are
intended for general public use, their control and navigational
behavior requires relatively low skill levels; attainable by most
people. Consequently this type of r/c toys offer a limited sense of
accomplishment to a user since the entertainment factor is
constrained mostly to magnitudes of speed and visual thematic
variations (such as colors and decorative ornamentation appealing
to human imagination).
In other occasions of prior art, creative variations of surface
roving r/c toy vehicles have been introduced, attempting to improve
the amusement factor, by use of mechanical adaptations for
performing various stunt maneuvers. For example some toy vehicles
were designed to be invertible, others were adapted for spinning in
place, yet others have adaptations for tumbling, or performing
wheelies, and so on. However, the recreational value of these toy
stunt vehicles lies in the assumption that a human being will
become amused by self-inflicted actions (initiating a stunt
maneuver and then watching it unfold). A user will, arguably, lose
interest sooner when handling a device that performs repeatedly a
staged action, instead of handling a device that imposes
spontaneous interaction, adaptation, and participation with actual
physical and environmental factors.
Accordingly it would be beneficial to have an educational surface
roving r/c toy vehicle designed with an inherent instability (such
as having a round profile prone to involuntary free rolling) that
the user would be called to manually compensate and thus be
continuously exposed to a plurality of unpredictable, spontaneous
(non-staged) environmental factors including gravity, inertial
forces, wind factors and ground surface morphology, that
dynamically affect the motion of the vehicle itself and the
navigation becomes a physical intuition challenge in its own
right.
SUMMARY OF THE INVENTION
Briefly stated, the present invention introduces a surface roving
vehicle which is remote controlled and features a pair of
omnidirectional wheels mounted on a single axle. This rover, while
in motion, has only two points of contact with the ground and it
has a round profile; by virtue of being simply a pair of wheels on
an axle.
The present rover has both axial and planar symmetry about the
center of its axle and about a perpendicular plane to the middle of
its axle respectively, and therefore enjoys the advantage of having
no sides (to tip over) and it does not require an up-down (or
similar) orientation in order for its wheels to remain in contact
with the ground. Consequently, this arrangement enables the present
rover to be stable only in the longitudinal direction of its axle,
whereas it can freely roll (and thus not stable) in the traverse
direction of its axle.
It is possible to balance the free-roll careening tendency and
further navigate the present rover on a desired path by having
direct control of two motion factors: Firstly, the ability to steer
by using onboard motors for rotating the omnidirectional wheels in
opposite directions. Secondly, the ability to speed in the
longitudinal direction by virtue of the omnidirectional wheels,
having actively driven (by onboard motors) secondary wheels
distributed around their perimeter, allowing the rover to translate
along its longitudinal axis.
An advantage of the present invention, particularly in a
recreational context, is the challenge it poses to the skill,
physical intuition, and coordination of the human operator
attempting to tackle the complexity of motion in view of the
free-rolling/careening effect (due to having only two points of
contacts with the ground).
The present invention rover also includes advanced dynamic
stability circuitry that communicates with onboard orientation
sensors and can act as a navigation assistant (autopilot) upon
user's request. Particularly, the dynamic stability circuitry is
programmed with intelligent algorithms, and controls the onboard
motors, to perform optimum stopping maneuvers (on behalf of the
user) when the user presses a stop button on the remote control
unit.
In other embodiments of the present invention (such as those
pertinent to terrain probing and exploration functions), the
dynamic stability circuitry has a more active role continuously
handling the low level detailed aspects of navigation and balance
(including compensation for the free-roll/careening effect) and the
user handles only higher level functions such as direction and
speed. The stability circuitry (in place of the user) is in direct
control of the onboard motors and uses feedback from the onboard
sensors to automatically find an optimum combination of inertial
roll, steering and thrust that will satisfy the direction and speed
requested by the user.
DRAWINGS--BRIEF DESCRIPTION OF THE FIGURES
FIG. 1 is a perspective view of a preferred embodiment of the
present invention.
FIG. 2 is a perspective view illustrating, in isolation, a frame,
axle and motor-housing of the preferred embodiment
FIG. 3 is a planar view depicting, in isolation, a wireless remote
controlled drive system of the preferred embodiment.
FIG. 4 is a perspective close-up view of an optomechanical roll
sensor.
FIG. 5 is an exploded perspective view of a left hub of the
preferred embodiment.
FIG. 6 is a perspective view of both a left and a right hub of the
preferred embodiment.
FIG. 7 is a cross sectional view of the preferred embodiment along
lines 7--7 from FIG. 1
FIG. 8 is a planar profile view of the preferred embodiment.
FIG. 9 is a planar profile view of a flexible shaft wheel.
FIG. 10 is a cross sectional view of the flexible shaft wheel along
lines 10--10 from FIG. 9.
FIG. 11 is a cross sectional slice of an alternate flexible shaft
wheel featuring a resilient portion and an elastomeric portion.
FIG. 12 is a perspective view of yet another embodiment of a
flexible shaft wheel having peripheral ground-traction features
FIG. 13A is a perspective view of an exemplary remote control unit
of the present invention.
FIG. 13B is a planar view of the remote control unit of FIG. 13A as
held by a user.
FIG. 14 is a diagrammatic plan view illustrating an automated
stop-sequence showing the position of the longitudinal axis of the
present invention rover during a succession of zigzag
maneuvers.
FIG. 15 is a planar view of another embodiment of the present
invention featuring a variation hub having traction features on its
surface.
FIG. 16 is a perspective scene featuring the preferred embodiment,
in motion, as guided by remote control and traveling on an incline
surface.
DETAILED DESCRIPTION OF THE INVENTION
The invention will next be described with respect to the figures.
The use of the words "left" and "right" denote the position of an
element within a figure relative to the left or right sides of the
drawing page. The figures are intended to be illustrative rather
than limiting. Numerous further variations, changes and
substitutions will occur to those skilled in the art without
departing from the spirit of the invention.
FIG. 1 is a summary perspective view of a preferred embodiment of
the present invention rover 100 in a complete, assembled,
functional form. Rover 100 is comprised by a left and a right
omnidirectional wheel 101L and 101R respectively and mounted on
each side of a frame 20. Wheels 101L,R share a common longitudinal
axis X and can rotate individually as indicated by arrows A1 and
A2. The rotation of wheels 101L,R enables a user to steer the rover
100 changing its direction. For example when wheel 100L rotates in
one direction and wheel 101R rotates in the opposite direction then
the rover 100 will turn about a vertical axis Z; as indicated by
arrow D3.
Also, as seen FIG. 1, each of omnidirectional wheels 101L,R
comprises a set of three secondary toroidal wheels 150L and 150R
respectively. Toroidal wheels 150L,R are distributed in a polar
array around the perimeter of omnidirectional wheels 101L,R.
Toroidal wheels 150L,R of the preferred embodiment are cylindrical
large diameter hollow flexible shafts (as discussed later in FIGS.
9 12), simplified to resemble coils bent to an arc and driven to
rotate flexibly as indicated by arrows B1 and B2; so as to provide
a constant velocity anywhere on their outer perimeter.
Consequently, when toroidal wheels 150L,R rotate synchronously, the
rover 100 moves along the longitudinal axis X in either direction
as indicated by arrow D1.
A user of the rover 100 has direct control of two motion factors,
namely i) the linear motion as shown by arrow D1 and ii) the
turning motion as shown by arrow D3. Furthermore, rover 100, being
substantially of round profile (and as discussed later in FIG. 8)
will also have an inertial tendency to careen (roll freely) as
indicated by arrow D2 and follow a path along an axis Y
perpendicular to the longitudinal axis X. Thus, the careening (free
rolling) effect constitutes a third motion factor and can only be
controlled indirectly by judicious combination of the previously
mentioned two direct motion factors. Also, the careening effect
depends on various parameters such as inertia due to change of
direction, or due to gravity forces depending on incline, or
imperfections of the surface where rover 100 is driven. This
behavior of having two directly controlled motion factors and a
third, unpredictable, free-rolling factor is done so by design so
as to challenge the skill and coordination of the user.
Although each set of toroidal wheels 150L,R is shown in FIG. 1 with
three individual wheels this should not be construed as a limiting
factor since it is possible to have arrays of less than or more
than three secondary wheels on each of omnidirectional wheels
101L,R. Also, in the preferred embodiment the sets of toroidal
wheels 150L,R may be identical or a mirror image of each other such
as for example the coils of toroidal wheels 150R may have reverse
twist compared to the coils of toroidal wheels 150L as seen in FIG.
1.
Now, in reference to FIG. 2, a perspective view of the frame 20 is
shown and discussed in isolation from the rest of the components of
rover 100 for illustrative purposes. As seen in FIG. 2 the frame 20
comprises a tubular axle 22 having a hollow center that forms a
cylindrical battery compartment cavity 38. Axle 22 has a pair of
threaded ends 40 that receive a pair of threaded caps 42 (seen in
FIG. 1) to close-off cavity 38. A cylindrical housing 24 is
situated midways on axle 22 and comprises an open ended tubular
shell 28 and a pair of housing caps 26 fitted on each end of shell
28 to define an enclosed volume 25. Tubular shell 28 is secured on
axle 22 by a pair of divider walls 30 separating the enclosed
volume 25 into two substantially equal parts. Divider walls 30 are
roughly quarter-disc shaped and diametrically opposed leaving a
pair of quarter openings 32 shaped to securely seat a pair of drive
motors M1 and M2 so as to be enclosed within housing 24 (the top
drive motor M1 is not shown in FIG. 2). Divider walls 30 may also
be thought of as the second and fourth quadrants of a solid disc
with the first and third quadrants being cut out to fit the drive
motors M1 and M2.
Housing 24 further encloses intelligent electronic circuitry
serving as a processing unit 33 which may be spread out (depending
on complexity) on a number of printed circuit boards (PCB) such as
PCBs 34a,b,c and d placed on either side of divider walls 30 (the
PCB 34d is not visible in FIG. 2). Although four PCBs are suggested
in FIG. 2, less than four may be used or more than four may be
stacked within housing 24 depending on complexity.
Moving on to FIG. 3, a perspective view is shown of a wireless
remote controlled drive system 50 of the present invention isolated
from the rest of the components of rover 100. Drive system 50
includes the PCBs 34a d (omitted from FIG. 3 to avoid illustration
clutter but shown in FIG. 2) and the drive motors M1 and M2 which
were previously mentioned in reference to FIG. 2.
In the preferred embodiment, drive system 50 further comprises a
left drivetrain 52L situated on the left of drive motors M1 and M2
and responsible for rotating the left omnidirectional wheel 101L
and a right drivetrain 52R situated on the right of drive motors M1
and M2 and responsible for rotating the right omnidirectional wheel
101R. Each of drivetrains 52L,R has a planetary gearing system 53
comprising a sun gear 54, a ring gear 56, a number of planetary
gears 66 (three planetary gears 66 are used in the preferred
embodiment) and a planetary gear carrier ring 58. The sun gears 54
(of both drivetrains 52L,R) have a sun gear portion 54a on one end,
a tubular shaft portion 54b in the middle, and a geared-end portion
54c on the other end. Similarly, ring gears 56 have a ring gear
portion 56a on one end, a tubular shaft portion 56b in the middle,
and a geared-end portion 56c on the other end. Tubular portions 56b
of ring gears 56 are concentric to, and rotate within, tubular
portions 54b of sun gears 54 about the longitudinal axis X. Each of
planetary gears 66 has a shaft 76 that is rotatably suspended
within corresponding holes 84a of planetary carrier ring 58 (as
best seen later in FIG. 5) so that planetary gears 66 are in a
polar array and mechanically connecting ring gear portions 56a to
sun gear portions 54a. Drive motor M1 has a shaft 78A fitted with a
drive motor gear 80a engaging both ring gears 56 (at the geared-end
portions 56c). Similarly, drive motor M2 has a shaft 78B fitted
with a drive motor gear 80b engaging both sun gears 54 (at the
geared-end portions 54c).
Each of drivetrains 52L,R also includes a number of spindle
transmissions 60 corresponding to the number of planetary gears 66
and each spindle transmission 60 comprises a spindle 68 having
keyed ends 70, a spindle output gear 62, a pair of spindle bearings
72 on either side of the output gear 62, and a connecting gear 64.
Each connecting gear 64 engages a corresponding one of planetary
gears 66 to the output gear 62 and has a shaft 74.
A person skilled in the art will recognize the planetary gearing
system 53 as a simple differential mechanism. The properties of
such differential mechanisms are advantageous in the present
invention in the sense that when the sun gear portions 54a and the
ring gear portions 56a are rotating in opposite directions with the
same circumferential velocity then the planetary gears 66 spin in
place about their shafts 76 on planetary carrier rings 58 (without
revolving around axis X) and therefore cause pure rotation of the
spindles 68. Conversely, when the sun gear portions 54a and the
ring gear portions 56a are rotating in the same direction with the
same circumferential velocity then the planetary gears 66 have a
zero spin about their shafts 76 but they revolve about axis X and
also cause the planetary carrier rings 58 to revolve about axis X;
and as a matter of fact the entire drivetrain 52L (or 52R) revolves
in unison, as a single piece, purely about axis X. Other
combinations of circumferential velocities between the ring gear
and sun gear portions 54a and 56a will cause hybrid scenarios of
the above mentioned pure conditions.
The left and right drivetrains 52L and 52R are a mirror image of
each other except the fact that each of connecting gears 64 on the
left drivetrain 52L engages the opposite end of its corresponding
planetary gear 66; as compared to the connecting gears 64 on the
right drivetrain 52R. Also, the ratio of a radius Ra to a radius Rb
of motor gears 80a and 80b respectively could be selected based on
equation 1
.times..times. ##EQU00001## where Ra1, Rb1, Ra2 and Rb2 are the
radii of tubular shaft portions 56b and 54b, as well as sun and
ring gear portions 56a and 54a respectively, so that equal
magnitudes of angular velocities of shafts 78A and 78B will produce
equal magnitudes of circumferential velocities on ring gear
portions 56a and sun gear portions 54a respectively.
In summary of the overall drivetrain functionality, when the
diametrically opposed motor gears 80a,b are both driven along the
same direction R (or direction L) then the tubular shaft portions
56b and 54b rotate in opposite directions, the carrier rings 58
remain stationary and all spindles (on both left and right side)
rotate in the same direction CC (or direction CL respectively).
Also, when one of the motor gears (80a or 80b) is driven along one
direction (R or L) while the other motor gear (80b or 80a) is
driven in the exact opposite direction (L or R) then all spindles
68 have zero spin and the entire left drivetrain 52L rotates about
axis X and in the opposite direction than the entire right
drivetrain 52R. Furthermore, the rotation of spindles 68 can be
varied in a continuum of speeds by applying equal amounts of
incremental change (positive or negative) in the rotational speed
of both motors M1 and M2. Conversely, the rotation of carrier rings
58 can be varied in a continuum of speeds by applying equal and
opposite amounts of incremental change in the rotational speed of
both motors M1 and M2.
Continuing the discussion of the wireless remote controlled drive
system 50, the processing unit 33 on PCBs 34a d serves as an
electronic brain for the present invention that intelligently
controls the speed of drive motors M1 and M2 in response to
wireless radio signals from a user-operated remote control unit 166
(as later seen in FIGS. 13A and 13B). The processing unit 33
includes a signal processing circuitry connected to an on-vehicle
antenna (not shown). The antenna may be a separate Wi-Fi antenna,
or preferably the tubular shell 28 itself (or any component of the
present invention rover 100) doubling its function as an antenna.
Alternatively other antenna forms may be used as for example
microstrip antennas integrated into the circuitry on PCBs 34a
d.
In the preferred embodiment of the present invention rover 100, the
drive system 50 further includes a number of sensors. Particularly,
PCBs 34a d include micro-sensors (not shown) such as those from
Freescale Semiconductor, Inc. (Austin Tex.); or Kionix, Inc.
(Ithaca, N.Y.); or PNI corporation (Santa Rosa Calif.); or
Honeywell International Inc (Morristown N.J.); or similar, for
sensing of:
a) tilt of axis X relative to horizontal,
b) direction of roll and rotational velocity of frame 20 (and thus
of the entire rover 100),
c) azimuth (north/south orientation of longitudinal axis X),
d) distance/range of rover 100 relative to the remote control unit
166.
In some embodiments of the present invention rover 100 (depending
on manufacturing complexities and cost), the processing unit 33
also receives feedback from an optomechanical roll sensor 116 (seen
in FIG. 4).
The roll sensor 116 as seen in FIG. 4 is well known in the art of
optomechanical sensors such as those used in computer mice, gaming
joysticks and the like. The roll sensor 116 can be located on
either end of axle 22 and comprises a slotted wheel 118 mounted on
a bearing 124. Wheel 118 has a plurality of radial slots 119 that
intermittently interrupt a pair of light beams between pairs of
light receptors 120 and light emitters 122 (best seen in FIG. 7)
situated on mounting brackets 123a and 123b. Roll sensor 116 also
has a semicircular gravity-follower weight 126 attached on the
slotted wheel 118. The weight 126 will always seek to rest at the
lowest position towards the earth thus causing wheel 118 to rotate
on bearing 124 as the rover 100 rolls/careens on a ground surface.
Sensor 116 is connected to the processing unit 33 on PCBs 34a d to
communicate direction of roll and rotational velocity of frame
20.
Now in reference to FIG. 5, an exploded perspective view is shown
illustrating a left hub 82L of omnidirectional wheel 101L. Hub 82L
comprises a toroidal channel 90 of roughly semicircular
cross-section creating a donut shaped hub opening 83 on its
underside of sufficient diameter to allow shaft portion 54b to pass
through without coming in contact. Channel 90 is interrupted by a
set of three identical channel dividers 92 each having sufficient
width to house a corresponding spindle transmission 60 while
leaving spindles 68 exposed from both ends. One side (the left) of
each of the channel dividers 92 has a slot opening 88 to allow the
gear teeth of connecting gear 64 (not shown in FIG. 5) to be
exposed. In between dividers 92, a corresponding one of left
toroidal wheels 150L (shown in FIG. 1) is fitted to be driven by
spindles 68 (in a manner which will become increasingly clear later
in this discussion). Adjacent to slots 88, on one side, is a set of
planetary-gear-shaft holes 84b corresponding to the gear-shaft
holes 84a on carrier ring 58. Also adjacent to slots 88 are
protrusions, or mounting features 86, integral to the underside of
channel 90 serving as attachment points for securing carrier ring
58 on hub 82L; with each of planetary gears 66 rotatable within a
corresponding pair of holes 84a and 84b (as seen in FIG. 6). The
left side of hub 82L is closed off by a hub cone cap 94 having an
opening 95 for axle 22 and cap 42.
FIG. 6 is a perspective view showing both left and right hubs 82L
and 82R. Hub 82L is shown with its cone cap 94 detached to reveal
the placement of carrier ring 58, planetary gears 66 and connecting
gears 64. The right hub 82R is a mirror image of left hub 82L with
the exception that each of the connecting gears 66 (of hub 82R) are
meshing on the opposite side of their corresponding planetary gear
66 (compared to connecting gears 66 of hub 82L as shown in FIG. 5)
and the mounting features 86 are located in a circumferential
shifted position accordingly. Carrier rings 58 are fixedly secured
on hubs 82L,R so that they act as a singe piece; and thus when the
carrier rings 58 rotate then the entirety of either hub 82L,R
rotates and consequently the entirety of either omnidirectional
wheel 101L,R rotates.
Now in reference to FIG. 7, a cross sectional view of the present
invention rover 100 is shown as taken by section lines 7--7 from
FIG. 1. In this view, all of the components of the preferred
embodiment can be seen in their assembled form, except the PCBs 34a
d (previously seen in FIG. 2) and the toroidal wheels 150L,R
(previously seen in FIG. 1). As it can be seen in FIG. 7 battery
compartment cavity 38 receives a number of batteries, or power
sources 108 (six are shown in FIG. 7). A divider 128 splits the
cavity 38 in two chambers separating the power sources 108 in two
columns. Each column of power sources 108 touches a lead 130 (of
divider 128) on one end and a lead 107 (on cap 42) on the other
end. The columns of power sources 108 are electrically connected in
parallel in the preferred embodiment and provide electrical energy
to the drive motors M1 and M2 as well as PCBs 34a d. However,
someone skilled in the art will note that a connection in-series is
also possible, or that the divider 128 can be omitted, or that the
first column of power sources 108 may be dedicated to one of drive
motors M1 and M2 and the second column to the other of drive motors
M1 and M2.
Additionally, the threaded caps 42 enable a user to access the
battery compartment cavity 38 from both ends of axle 22 so as to
replenish power sources 108 when needed. Threaded caps 42, comprise
a rigid threaded portion 104 and a flexible, or cushioning, domed
portion 106. Domed portion 106 defines a cavity 102 and serves as a
bumper or cushion to absorb shock when rover 100 is driven
carelessly and bumps or pushes longitudinally against obstacles in
its course. Also, caps 42 form the tips, and cone caps 94 form the
base, of a pyramid shape (as best seen in FIG. 1) helping to
prevent any possibility of rover 100 balancing vertically with both
wheels 101L,R off the ground. Hubs 82L and 82R as well as sun gears
54 and ring gears 56 are rotatably fitted in coaxial layers around
axle 22. Particularly, each of tubular shaft portions 56b is placed
coaxially over axle 22, on each side of housing 24, and suspended
on bearings 110a and 110b. Each of tubular shaft portions 54b is
placed coaxially over the tubular shaft portion 56b of the
corresponding ring gear 56, and is suspended on bearing 112.
Finally, each of hubs 82L and 82R is suspended on bearings 111b and
111a and has the shaft portion 54b of the corresponding sun gear 54
passing through opening 83. Bearings 111a and 112 are seated on
corresponding bearing seats 44b and 44a (best seen in FIG. 2)
respectively, of housing caps 26, and thus they are intimate to
frame 20. Bearings 110a, 110b and 111b are seated on axle 22 and
thus are also intimate to frame 20.
Now in reference to FIG. 8, a profile view of the rover 100 is
shown. In this figure the right toroidal wheels 150R are aligned
directly behind wheels 150L and thus hidden from view. Rover 100
preferably comprises three toroidal wheels 150L (as well as three
wheels 150R) each being elastically bent (during assembly) into an
arc and situated in between a pair of consecutive dividers 92; thus
wheels 150L,R define a substantially circular outer boundary
profile for rover 100. Dividers 92 cause the ends of toroidal
wheels 150L,R to be separated by a gap G1. Furthermore, each one of
bent toroidal wheels 150L,R has a coil outer pitch G2. In the
preferred embodiment, gap G1 is desired to be as small as possible
or better yet equal to the coil outer pitch G2. Also the symmetry
and shape of hubs 82L,R as well as frame 20 and the distribution of
internal components, is such that the weight of rover 100 is
uniformly balanced in both radial and longitudinal directions and
thus the center of gravity coincides with axis X and particularly
with point CG (as best seen in FIG. 7). The round profile of the
present invention rover 100 is an important deliberate feature
introducing an element of unpredictable inertial roll during use,
where the user's skill, physical intuition, and eye-hand
coordination are put to a test.
Moving on to FIGS. 9 and 10, a profile view and a cross-sectional
view respectively are shown illustrating one of the right toroidal
wheels 150R (the rest of toroidal wheels 150L,R are similar in
construction). The toroidal wheel 150R is a simplified cylindrical
flexible shaft which, in this case, comprises three identical coils
152a, 152b and 152c. Each of coils 152a c has an elongated
flattened cross-section (as seen in FIG. 10) and is connected at
each end to a rim 155 of a hub 154. More specifically, the ends of
coils 152a c are attached to, and distributed in a polar array
around, rims 155 (best seen in FIG. 9). Each of hubs 154 has a
keyed-opening 156 that is complementary to the keyed ends 70 of
spindles 68 (shown previously in FIG. 3).
During assembly, each of the toroidal wheels 150L and 150R are
placed within channel 90 of hubs 82L,R and is elastically deflected
so that each end receives a spindle 68 into the keyed-opening 156
of the corresponding hub 154. The inherent tendency of the
deflected coils 152a c to return to their natural position
generates an internal force that pushes the coils firmly against
spindles 68 and thus keeping wheels 150L,R from detaching during
normal use. However, wheels 150L,R can be easily detached by a user
and replaced if needed by simply forcing the ends of toroidal
wheels 150L,R out of their spindles 68. Although the wheel 150R
shown in FIGS. 9 and 10 has three coils 152a c, more than three or
less than three coils may be used. Also, other embodiments of coils
152a c are possible that have different cross sectional shapes,
such as (but not limited to) circular, square, elliptic and the
like. Coils 152a c are resilient (spring-like) but have sufficient
stiffness so that wheels 150L,R exhibit minimal deflection due to
the weight of rover 100, or due to acceleration/deceleration forces
as well as bumps encountered on the terrain. In other words the
toroidal wheels are so stiff that any deflection during normal use
will not cause them to come in contact with the surface of channel
90 except in the most severe circumstances.
Furthermore, a person skilled in the art would point out that
flexible shaft drives, are meant to be quite efficient (nearly 90
95% efficiency) and thus one of the desirable characteristics is to
have the least possible internal friction for minimum loss of
rotational energy. Consequently, the materials chosen for the coils
152a c of the present invention rover 100 are preferably of low
internal friction so as to realize low bending stiffness
(consistent with the art of flexible shaft drives) but at the same
time have surface characteristics that provide enough traction so
as to bestow rover 100 with a meaningful grip on a variety of
terrain types.
An alternate embodiment coil 158 designed for high traction is
shown in cross-section in FIG. 11. Instead of single material coils
152a d, the coil 158 comprises two layers. Particularly, coil 158
comprises an elastomeric layer 159a, as a shell, and a resilient
core 159c for added stiffness. Layer 159a also has a ground
engaging outer portion 159b, with more material, so as to
compensate for normal wear and be suitable for traction on a
variety of terrain types.
FIG. 12 is a perspective view of another embodiment showing an
alternate toroidal wheel 160 comprising a single coil 162. The coil
162 has a similar cross-sectional profile as coils 152a c and has a
plurality of traction features 164 distributed along its outer
edge. Traction features 164 are preferably of elastomeric material
and can be of any suitable shape (including spherical bumps as
shown in FIG. 12) that is known to enhance ground traction.
Now in reference to FIGS. 13A and 13B, an exemplary remote control
unit 166 is shown for wireless remote manipulation of the present
invention rover 100. Unit 166 comprises a steering knob control 168
positioned to rotate on a vertical axis. Steering knob 168 has
upper and lower rotary knobs 168A and 168B which are mechanically
linked by sharing a common shaft and thus rotate as a single body.
Lower rotary knob 168B is ergonomically located so as to be
accessible by a users index finger 179 thus enabling single handed
operation as seen in FIG. 13B. Upper rotary knob 168A may be used
for operating the control unit 166 with both hands, as for example,
with a users left hand 176L holding the unit 166 while a users
right hand is used to handle steering, via the upper rotary knob
168A, or vice-versa.
In addition, unit 166 also comprises a partially exposed thrust
control knob 170 rotatable about a horizontal axis and
ergonomically positioned for access by a user's thumb 178. In the
preferred embodiment of the present invention rover 100, the left
and right hubs 82L,R are color coded so that hub 82L has one vivid
color and hub 82R has another vivid color mutually contrasting so
that each of hubs 82L,R can be visually referenced from a distance.
Furthermore, remote control unit 166 has instructive color-coded
reference surface markings 174a and 174b (shown as forward and
rearward arrows in FIG. 13A) in the vicinity of thrust control knob
170. Each of markings 174a and 174b have a matching color that
corresponds to each of hubs 82L and 82R respectively. This color
coding arrangement acts as a visual aid, during use, so as to
minimize potential confusion as to which direction the rover 100
will move in response to thrust control knob 170.
Remote control unit 166 communicates steering and thrust signals to
be interpreted by the processing unit 33 of rover 100. The
processing unit 33 will, in turn, regulates the rotational speed of
drive motors M1 and M2 accordingly. Particularly, the thrust
control knob 170 results in thrust signals that are purely
affecting the rotation of spindles 68 and thus toroidal wheels
150L,R so as to drive the rover 100 in forward, or reverse,
direction along longitudinal axis X. The steering knob 168 results
in steering signals that are purely affecting the rotation of hubs
82L,R and thus omnidirectional wheels 101L,R in opposite directions
so as to turn the longitudinal axis X of rover 100.
More specifically, the processing unit 33 translates the steering
signals into equal and opposite amounts of change on the rotational
speed of motors M1 and M2 (such as a positive speed increment on
motor M1 and a negative speed increment on motor M2, or
vice-versa). Also, the processing unit 33 translates the thrust
signals into equal amounts of change on the rotational speed of
both motors M1 and M2 (such as a negative speed increment on both
motors M1 and M2 or, conversely, a positive speed increment on both
motors M1 and M2). Furthermore, when a combination of both steering
and thrust signals is sent by the user, then the signals are
interpreted, by the processing unit 33, so that each of motors M1
and M2 is controlled by the algebraic sum of the constituent motor
speed requirements (from the steering-thrust combination signal).
For example (in one of many possible scenarios): a) if the steering
aspect (of the steering-thrust combination signal) requires a
positive speed increment (t1) on motor M1 and a negative speed
increment (-s1) on motor M2, and b) if the thrust aspect (of the
steering-thrust combination signal) requires a positive speed
increment (t1) on both motors M1 and M2, c) then the processing
unit 33 will perform an algebraic sum of the requirements of
conditions (a) and (b) so that the speeds of motors M1 and M2 are
adjusted by the amount of delta1=s1+t1 (for motor M1) and the
amount of delta2=-s1+t1 (for motor M2). Furthermore, continuing the
above example, if the value t1 is less than the value s1 (thus
delta2=-s1+t1<0) and the motor M2 was initially at rest, then
motor M2 will end up being driven in reverse (a speed of
delta2=-s1+t1). If the value s1 equals the value t1 and the motor
M2 was initially at rest, then motor M2 will remain at rest (since
in this case delta2=-s1+t1=0) while the speed of motor M1 will be
further increased by the amount of delta1=s1+t1
The remote control unit 166 further comprises an emergency stop
button 172 which serves to override all other user controls,
including steering and thrust control knobs 168 and 170, and sends
a distress signal to rover 100; so as to initiate an automatic stop
sequence. The rover 100 further comprises a dynamic stability
circuitry, included in the processing unit 33, which takes over
control of motors M1 and M2 upon reception of the distress signal.
The stability circuitry continuously monitors feedback from the
onboard sensors of rover 100 and when the user presses the
emergency stop button 172 the stability circuitry takes into
account parameters such as current inertial-rolling speed and
direction as well as inclination of axis X relative to the true
horizon and makes use of built-in intelligence to bring the rover
to a stop in the shortest possible distance E1 (shown in FIG.
14).
For example, (and in reference to FIG. 14) rover 100 has a starting
inertial rolling speed and direction SP1 indicated as a phase 1.
The user decides that it is not possible to manually stop the
vehicle in time before an obstacle is hit and thus the user presses
the emergency stop button 172. Upon pressing button 172, the
dynamic stability circuitry takes over and the rover 100 enters a
phase 2. During phase 2, and depending on the rolling speed SP1,
the rover 100 will engage into an automated first zigzag maneuver
190a. A zigzag maneuver is a pair of successive abrupt convex-angle
turns in opposite directions; with each turn causing the inertial
rolling speed SP1, of rover 100, to be reduced by a small amount.
The degree of turning during each convex-angle turn is
intelligently determined by algorithms programmed in the dynamic
stability circuitry ensuring that the centrifugal forces generated
will not cause rover 100 to loose traction (and/or start tumbling).
Depending on the remaining speed at the end of the first maneuver
190a, further zigzag maneuvers 190b, 190c, and so on, may be
applied successively (with increasing degree of turning) until a
last maneuver 190n brings the vehicle to a safe speed so as to
enter a phase 3. At phase 3 the rover 100 can finally turn and
align its longitudinal axis X in a direction that counteracts any
further rolling tendency due to the incline of the ground surface
and thus reaching a dynamic rest position Q. The rover 100 will
actively maintain the rest position Q (compensating for external
distracting factors such as wind, ground surface vibrations, and
the like) for as long as the user keeps holding the stop button
172.
Although a zigzag stopping maneuver has been shown in FIG. 14 in
association with the present invention rover 100, other stopping
movements are also possible. For example, in another embodiment,
rover 100 utilizes a different set of algorithms causing it to
follow a spiraling path (instead of zigzag), or a combination of
zigzag and spiraling paths, to achieve an optimum minimal stopping
distance E1.
In another embodiment of the present invention rover 100, the
dynamic stability circuitry on PCBs 34a d is also programmed with
self-preservation algorithms that will initiate the stopping
sequence (as described in the above example) in situations other
than in response to the stop button 172. Particularly, the stop
sequence may also be initiated if rover 100 senses that the energy
from the power sources 108 has diminished bellow a specified
threshold or when rover 100 senses that wireless communication with
the remote control unit 166 is being interrupted. In those
situations rover 100 will maintain the dynamic stop position and
the user is then alerted by a combination of visible and audible
signals from the remote control unit 166 so that rover 100 can be
recovered by hand.
In yet another embodiment of the present invention rover 100, the
remote control unit 166 may also allow selection of various
difficulty levels (such as novice level, or advanced level and the
like). For example, if the novice difficulty level is selected,
then the dynamic stability circuitry on PCBs 34a d will switch to
the most advanced set of algorithms and the signals from steering
control knob 168 will be interpreted as direction signals (by using
feedback from the onboard sensors such as magnetic azimuth
sensors/compass). The dynamic stability circuitry will become
actively engaged assisting the user during navigation. The user
merely points the desired direction, by turn of steering control
knob 168, and the dynamic stability circuitry will automatically
pilot, by application of necessary course-corrective actions,
finding an optimum combination of inertial roll, steering and
thrust that will satisfy the direction and speed requested by the
user to guide the rover 100.
Now in reference to FIG. 15, a planar view of an alternate
embodiment rover 100a is shown comprising a variation hub cone cup
96 on each of hubs 82L and 82R. Each of cone cups 96 includes a
plurality of indentations 98 providing surface variation and
roughness to the cone cup 96 so as to aid traction in certain
special circumstances. Particularly, in the event that rover 100a
is driven on rough or obstacle ridden terrain there is possibility
for a situation where rover 100a is suspended from both ends on
ground objects such as masses 140a and 140b where the wheels 150L,R
have both lost contact with a ground surface 142. The user is able
to negotiate the situation by applying a steering action, where
hubs 82L,R counter-rotate, so as to roll on masses 140a,b in
opposite directions and escape the predicament. Indentations 98
include steps, or edges 98a, on the surface of cone cups 96 that
help improve grip against masses 140a and 140b.
Finally, in reference to FIG. 16, a perspective view of the present
invention rover 100 is shown in operation being driven on an
incline surface 180 in response to signals from the remote control
unit 166 held by a user's right hand 176R. Rover 100 is designed
deliberately with a capability to challenge the skill, physical
intuition and coordination of a user. Rover 100 may be driven on a
track 194 having various types of adjacent surfaces with varying
degrees of inclination including a horizontal surface 182; where
the user attempts to navigate the rover 100 precisely through sets
of goals 192. The complexity and challenge of operation is
particularly evident when rover 100 is driven on an incline surface
where the involuntary inertial rolling aspect is also affected by
the forces of gravity. Particularly, when rover 100 is driven on
incline surface 180 at an angle .alpha. to the horizontal the user
will have to manually determine a velocity component vector V1 (via
thrust control knob 170) as well as the optimum steering
orientation (handled via steering knob 168) so that when combined
with an existing velocity component vector V2 (due to involuntary
inertial and gravity rolling) rover 100 will have a velocity
resultant vector V3 and thus follow a desired path P leading toward
goals 192. As it can be seen in FIG. 16 and contrary to expectation
the longitudinal axis of rover 100 and the path P of movement need
not necessarily coincide.
TABLE-US-00001 DRAWINGS - REFERENCE NUMERALS 20 frame 22 axle 24
housing 25 volume 26 housing caps 28 tubular shell 30 divider walls
32 openings 33 processing unit 34a d printed circuit boards (PCB)
M1 first drive motor M2 second drive motor 38 battery compartment
cavity 40 threaded end 42 threaded cap 44a,b bearing seat 50
wireless remote controlled drive system 52L left drivetrain 52R
right drivetrain 53 planetary gearing system 54 sun gear 54a sun
gear portion 54b tubular shaft portion 54c geared-end portion 56
ring gear 56a ring gear portion 56b tubular shaft portion 56c
geared-end portion 58 planetary gear carrier ring 60 spindle
transmission 62 spindle output gear 64 connecting gear 66 planetary
gear 68 spindle 70 keyed end 72 spindle bearing 74 connecting-gear
shaft 76 planetary gear shaft 78a,b shaft 80a,b drive motor gear
82L left hub 82R right hub 83 opening 84a,b planetary-gear-shaft
holes 86 mounting features 88 slot openings 90 toroidal channel 92
channel dividers 94 hub cone cap 95 opening 96 hub cone cup 98
indentations 98a edges 100 rover 100a rover 101L left
omnidirectional wheel 101R right omnidirectional wheel 102 cavity
104 threaded portion 106 domed portion 107 leads 108 power sources
110a bearings 110b bearings 111a bearing 111b bearing 112 bearings
116 optomechanical roll sensor 118 slotted wheel 119 slots 120
light receptor 122 light emitter 123a first mounting bracket 123b
second mounting bracket 124 bearing 126 gravity-follower weight 128
divider 130 leads 140a,b mass 142 ground 150L left toroidal wheels
150R right toroidal wheels 152a c coils 154 hub 155 rim 156
keyed-opening 158 coil 159a elastomeric layer 159b ground engaging
portion 159c resilient core 160 toroidal wheel 162 coil 164
traction features 166 remote control unit 168 steering knob 168a
upper rotary knob 168b lower rotary knob 170 thrust control knob
172 emergency stop button 174a markings 174b markings 176L left
hand 176R right hand 178 thumb 179 index finger 180 incline surface
182 horizontal surface 190a first zigzag maneuver 190b second
zigzag maneuver 190c third zigzag maneuver 190n final zigzag
maneuver 192 goal 194 track .alpha. angle A1 arrow A2 arrow B1
arrow B2 arrow CC direction CG center of gravity point CL direction
D1 arrow D2 arrow D3 arrow E1 distance G1 gap G2 coil outer pitch L
direction P path Q rest position R direction Ra radius Ra1 radius
Ra2 radius Rb radius Rb1 radius Rb2 radius SP1 inertial rolling
speed V1 velocity component vector V2 velocity component vector V3
velocity resultant vector X longitudinal axis Y axis Z axis
* * * * *